What least number should be deducted from 9847 so that it gives remainder 9, 8, 7, 6.... 1 when divided by 10, 9, 8,7 ..... 2 respectively?

Options

A. A)2286
B. B)2317
C. C)2288
D. D)2243

Answer

To solve the problem of finding the least number that should be deducted from 9847 so that it gives remainders 9, 8, 7, 6, ..., 1 when divided by 10, 9, 8, 7, ..., 2 respectively, we can follow these steps:

### Step 1: Understand the Remainders
We need to find a number \( x \) such that:
- \( 9847 - x \equiv 9 \mod 10 \)
- \( 9847 - x \equiv 8 \mod 9 \)
- \( 9847 - x \equiv 7 \mod 8 \)
- \( 9847 - x \equiv 6 \mod 7 \)
- \( 9847 - x \equiv 5 \mod 6 \)
- \( 9847 - x \equiv 4 \mod 5 \)
- \( 9847 - x \equiv 3 \mod 4 \)
- \( 9847 - x \equiv 2 \mod 3 \)
- \( 9847 - x \equiv 1 \mod 2 \)

### Step 2: Set Up the Equations
From the above conditions, we can rewrite them as:
- \( x \equiv 8 \mod 10 \)
- \( x \equiv 7 \mod 9 \)
- \( x \equiv 6 \mod 8 \)
- \( x \equiv 5 \mod 7 \)
- \( x \equiv 4 \mod 6 \)
- \( x \equiv 3 \mod 5 \)
- \( x \equiv 2 \mod 4 \)
- \( x \equiv 1 \mod 3 \)
- \( x \equiv 0 \mod 2 \)

### Step 3: Solve the Congruences
We can solve these congruences starting from the largest modulus down to the smallest.

1. From \( x \equiv 8 \mod 10 \), we can express \( x \) as:
\[ x = 10k + 8 \quad \text{for some integer } k \]

2. Substitute \( x \) into the next equation \( x \equiv 7 \mod 9 \):
\[ 10k + 8 \equiv 7 \mod 9 \implies k \equiv 8 \mod 9 \implies k = 9m + 8 \]
Therefore,
\[ x = 10(9m + 8) + 8 = 90m + 80 + 8 = 90m + 88 \]

3. Now substitute into \( x \equiv 6 \mod 8 \):
\[ 90m + 88 \equiv 6 \mod 8 \implies 2m + 0 \equiv 6 \mod 8 \implies 2m \equiv 6 \mod 8 \implies m \equiv 3 \mod 4 \]
So,
\[ m = 4n + 3 \]
Thus,
\[ x = 90(4n + 3) + 88 = 360n + 270 + 88 = 360n + 358 \]

4. Continuing this process for the remaining congruences will eventually yield a specific value for \( x \).

### Step 4: Find the Least Number to Deduct
To find the least number to deduct from 9847:
\[ x = 358 \quad \text{(for } n = 0\text{)} \]
Thus, the least number to be deducted from 9847 is:
\[ 9847 - 358 = 9489 \]

### Final Answer
The least number that should be deducted from 9847 is **358**.